15 Simple Harmonic Motion
Lots of things vibrate or oscillate. A vibrating tuning fork, a moving child’s playground swing, and the loudspeaker in a radio are all examples of physical vibrations. There are also electrical and acoustical vibrations, such as radio signals and the sound you get when blowing across the top of an open bottle.
One simple system that vibrates is a mass hanging from a spring. The force applied by an ideal spring is proportional to how much it is stretched or compressed. Given this force behavior, the up and down motion of the mass is called simple harmonic and the position can be modeled with
in Class we have used the equation: y = A sin(wt + d) but recall that 2pf = w
In this equation, y is the vertical displacement from the equilibrium position, A is the amplitude of the motion, f is the frequency of the oscillation, t is the time, and f is a phase constant. This experiment will clarify each of these terms.
Figure 1
objectives
· Measure the position and velocity as a function of time for an oscillating mass and spring system.
· Compare the observed motion of a mass and spring system to a mathematical model of simple harmonic motion.
· Determine the amplitude, period, and phase constant of the observed simple harmonic motion.
Materials
Power Macintosh or Windows PC 
ring stand, rod, and clamp 
LabPro or Universal Lab Interface Logger Pro Vernier Motion Detector 
spring, with a spring constant of approximately 10 N/m  We will use the springs that came with the PASCO carts twist ties 
50.0g and 100g masses (or just use two 50.0g masses) 
wire basket 
PreLab Questions
1. The motion detector is plugged into DIG/SONIC 1 or DIG/SONIC2?
2. How far
below the mass should you place the motion detector?
BE CAREFUL NOT TO STRIKE THE MOTION DETECTOR WITH
THE MASS!
3.
How do you measure the equilibrium distance for the mass on the spring?
4.
When you set the mass/spring in motion, do you lift the mass UP 5 cm and
then release it or DOWN 5cm and then release it?
5. When the mass/spring is in motion, should it be swinging to and fro? eg. side to side?
Procedure:
1. Attach
the spring to a ring stand and hang the mass
from the ring as shown in Figure 1. Securely fasten the 50g mass to the
spring and the spring to the rod.
I advise using tape to secure the spring to the ring stand, and the
masses to the spring.
2. Connect the Motion Detector to DIG/SONIC 2 of the LabPro
3. Place the Motion Detector at least 7075 cm below the mass. Make sure there are no objects near the path between the detector and mass, such as a table edge. Place the wire basket over the Motion Detector to protect it.
4. Open the LoggerPro file in the Experiment 15 folder of Physics with Computers. Graphs of distance vs. time and velocity vs. time are displayed.
5. Make a preliminary run to make sure things are set up correctly. Lift the mass upward a few centimeters and release. The mass should oscillate along a vertical line only. Click to begin data collection.
6.
After 10 s, data collection will stop. The position graph should show a
clean sinusoidal curve. If it has flat regions or spikes,
reposition the Motion Detector and try again. I
advise hanging the weights and spring off the end of the desk and supporting the
ring stand with a book. If
your data is not reflecting the motion of the spring you may need to recalibrate
the detector.
To
Calibrate:
a.
Place an object aproximatly one centemeter from the top of the motion
detector face and click “zero” in the “experiment” menu in Logger Pro.
b.
Now, click “calibrate” from the experiment menu. Select the detector
and press the button “calibrate.” Hold
a card at 0.5 m above the detector; use 0.5 m as the first value, and press
“keep.” Do the same for 1 m in
the second field.
7. Compare the position graph to your sketched prediction in the Preliminary Questions. How are the graphs similar? How are they different? Also, compare the velocity graph to your prediction.
8. Measure the equilibrium position of the 50g mass. Do this by allowing the mass to hang free and at rest. Click to begin data collection. After collection stops, click the statistics button, , (or go under the pull down menu "analyze" which has "statistics" under it) to determine the average distance from the detector. Record this position (y0) in the data table. It might read "zero" if you calibrated the detector such that the original height is read to be zero distance away from the detector.
9. Now lift the mass upward about 5 cm and release it. The mass should oscillate along a vertical line only. Let the mass bounce for a while (56 seconds) to make sure it is oscillating straight. Then, Click to collect data. Examine the graphs. The pattern you are observing is characteristic of simple harmonic motion.
10. Using the distance graph, measure the time interval between maximum positions. Here's how you do that: Click and drag the mouse across two maximums of the sine curve. When you have done this, go up to "Analyze" and select "examine" this should tell you the times between the two maximums (if you still don't get it........drag the mouse to the first maximum to record its time and then subtract that from the time over the second maximum). This is the period, T, of the motion (number of seconds between one oscillation). The frequency, f (number of oscillations per second), is the reciprocal of the period, f = 1/T. Based on your period measurement, calculate the frequency. Record the period and frequency of this motion in the data table.
11. The amplitude, A, of simple harmonic motion is the maximum distance from the equilibrium position. Estimate values for the amplitude from your position graph. Enter the values in your data table. Click on the Examine button, , once again to turn off the examine mode.
12. Repeat Steps 8 – 11 with the same 50g mass, moving with a larger amplitude than in the first run.
13. Change the mass to 100 g and repeat Steps 8 – 11. Use an amplitude of about 5 cm. Keep a good run made with this 100g mass on the screen. You will use it for several of the Analysis questions. Here's what I want from you as far as graphs are concerned: I want a graph of the 50g mass and then "STORE LATEST RUN" and plot a graph of the 100 g mass. This graph must be pasted right here. (Let me be perfectly clear: You must turn in a graph which has the run of a 50 g mass and a 100 g mass both on the same graph! right here, above the data table.)
12
1 Data Table
Run 
Mass 
y0 
A 
T 
f 

(g) 
(cm) 
(cm) 
(s) 
(Hz) 
1 





2 





3 





PostLab Questions:
1. View the graphs of the last run on the screen. Compare the position vs. time and the velocity vs. time graphs. How are they the same? How are they different? Paste a copy of the graph you are using to answer this question here as well!
2. Turn on the Examine mode by clicking the Examine button, . Move the mouse cursor back and forth across the graph to view the data values for the last run on the screen. Where is the mass when the velocity is zero? Where is the mass when the velocity is greatest? Paste a copy of the graph you are using to answer this question here as well!
3. Does the frequency, f, appear to depend on the amplitude of the motion? Do you have enough data to draw a firm conclusion? (no mass change, just amplitude change!)
4. Does the frequency, f, appear to depend on the mass used? Did it change much in your tests?
5. You can compare your experimental data to the sinusoidal function model using the Manual Fit feature of Logger Pro. Try it with your 100g data. The model equation in the introduction, which is similar to the one in many textbooks, gives the displacement from equilibrium. However, your Motion Detector reports the distance from the detector. To compare the model to your data, add the equilibrium distance to the model; that is, use
where y0 represents the equilibrium distance. (normally, this would be zero, but your detector might not see it that way. We try to zero out the detector, but often you get a value that is just above or just below zero.) Think of f as your phase shift in the horizontal direction and y_{o} as your phase change in the vertical direction and it should be clearer to you.
The following letters (ag) are intended to help you perform a manual fit of your Sine wave. At the end of letter g, you are going to paste your manual fitted graph AND show me the equation you used to fit it.
a. Click once on the distance graph to select it.
b. Choose Manual Fit from the Analyze menu.
c. Select the Sine function from the General Equation dropdown menu. (you might have to scroll down a little bit if you at first don't see the Sine function.)
d. The Sine equation is of the form y=A*sin(B*X+C) + D. Compare this to the form of the equation above to match variables; e.g., f corresponds to C, and 2pf corresponds to B. and X is time. A of course is the maximum amplitude in the y direction.
e. Adjust the values for A, B and D to reflect your values for A, f and y0. You can either enter the values directly in the dialog box or you can use the up and down arrows to adjust the values.
f. The phase parameter f is called the phase constant and is used to adjust the y value reported by the model at t = 0 so that it matches your data. Since data collection did not necessarily begin when the mass was at the equilibrium position, f is needed to achieve a good match.
g. The optimum value for f will be between 0 and 2p. (you kind of have to guess at the value for f . If the y_{o} is close to zero, then your value for f is close to zero. If y_{o} is close to the first maximum, then your value for f is closer to p/2....etc. You kind of just have to guess a little bit and see what happens.) Find a value for f that makes the model come as close as possible to the data of your 100 g experiment. You may also want to adjust y0, A, and f to improve the fit. Paste your graph here and Write down the equation that best matches your data.
6. Predict what would happen to the plot of the model if you doubled the parameter for A by sketching both the current model and the new model with doubled A. Now double the parameter for A in the manual fit dialog box to compare to your prediction. Paste your graph here: (no equation needed, I can double A in my head)
7. Similarly, predict how the model plot would change if you doubled f, and then check by modifying the model definition. Paste your graph here: (no equation needed, I can double f in my head)
8. When you look at the graphs you plotted, which ones had more oscillations per time interval? The 50g mass or the 100 g mass?
9. Write the equation for determining period of a spring if you know the mass and the spring constant (k). Does this agree with the results you got to question #8?
What is due for this lab:
1. Prelab Questions
2. Graph of the 50g mass and 100 g mass right above the data table.
3. Data Table
4. Graphs created on LOGGER PRO. As indicated in the postlab questions section. Paste them right there on the question so that I can see the graph while reading your postlab answers
5. PostLab Questions (see "what is due" #4 above)  please send me the questions with the answers and the graphs. Don't just send me graphs and answers!
Send it to me on email as "gy15SHM" where "gy" are the initials of the person writing this lab.
gyyoung@capousd.org